2024/03/12 by Colin Defant, Defant, Colin, Leigh Foster +7
Computer Science · Materials Science · #05A15 #05B45 #Combinatorics (math.CO) #FOS: Mathematics #Quasicrystal Structures and Properties #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2403.07663
openalex publication_date 2024/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Defant, Li, Propp, and Young recently resolved two enumerative conjectures of Propp concerning the tilings of regions in the hexagonal grid called benzels using two types of prototiles called stones and bones (with varying constraints on allowed orientations of the tiles). Their primary tool, a bijection called compression that converts certain k-ribbon tilings to (k-1)-ribbon tilings, allowed them to reduce their problems to the enumeration of dimers (i.e., perfect matchings) of certain graphs. We present a generalized version of compression that no longer relies on the perspective of partitions and skew shapes. Using this strengthened tool, we resolve three more of Propp's conjectures and recast several others as problems about perfect matchings.